PHYSICA MACEDONICA

INSTITUTE OF PHYSICS
Sts. Cyril and Methodius University, Skopje, Macedonia
: - (), - , - , - , - . EDITORIAL BOARD: D-r Blagoja Veljanoski (president), d-r Nace Stojanov, d-r Nenad Novkovski, d-r Olga Galbova, d-r Dragan Jakimovski. : - , , , . 162, 1000 , ADRESS: Sts. Cyril and Methodius University, Faculty of Natural Sciences and Mathematics, Institute of Physics, P.O.Box 162, 1000 Skopje, Republic of Macedonia The journal computer preparation is done at the Institute of Physics.
Physica Macedonica Vol. 62 pp. 1 - 47 Skopje 2016
C O N T E N T S
1. B. Radiša, M. Mitrovi and B. Misailovi ................................................................................. 1 THE EFFECT OF SOLUTION PRE-HISTORYON CHIRALITY OF SODIUM CHLORATE CRYSTALS FROM AQUEOUS SOLUTIONS
2. S. Rendevski, I. Aliu and N. Mahmudi ...................................................................................... 7 NORMALIZED INTEGRAL FOURIER TRANSFORM AMPLITUDE FOR FOOD QUALITY DETERMINATION BY ACOUSTIC IMPULSE RESPONSE METHOD
3. D. Krstovska and B. Mitreska ................................................................................................. 13 NONLINEAR ULTRASOUND WAVE GENERATION IN QUASI-TWO DIMENSIONAL ORGANIC CONDUCTORS
4. O. Atanackovi ........................................................................................................................ 22 ON TWO EXTREMELY FAST METHODS FOR THE SOLUTION OF THE NON-LTE LINE FORMATION PROBLEM
5. Luljeta Disha, Defrim Bulku and Antoneta Deda ................................................................... 34 METHOD OF ABSOLUTE GRAVITY MEASUREMENT AND ITS INFLUENCE IN MASS DETERMINATION
6. N. Mahmudi and S. Rendevski ............................................................................................... 41 GAMMA RADIATION SYNTHESIS OF POLYMERS AND CONTROL OF THE MECHANICHAL PROPERTIES
Physica Macedonica 62, (2016) p. 1-6
THE EFFECT OF SOLUTION PRE-HISTORYON CHIRALITY OF SODIUM CHLORATE CRYSTALS FROM AQUEOUS
SOLUTIONS
University of Belgrade-Faculty of Physics, Studentski trg 12, 11000 Belgrade, Serbia
Abstract. The phenomenon of spontaneous chiral symmetry breaking attracts the attention in physics, chemistry and biology. Sodium chlorate crystals are an interesting example of optically active system. In this paper results of investigations of effect of growth history on sodium chlorate crystals chirality will be presented. Crystals were nucleated and grown at T = (28.00±0.02)°C from aqueous solution saturated at T0 = (31.0±0.1)°C. Prior to nucleation the solution temperature firstly increased to Tinc, and then slowly decreased to growth temperature. Obtained results showed that solution pre-history induced the increase in the population of only one of enantiomers, i.e. D-crystals dominated. Results indicate that solution pre-history might affect chiral symmetry breaking in crystallization of sodium chlorate crystals even achieving crystals of chiral purity.
PACS: 81.10.Aj, 81.10.Dn
1. INTRODUCTION
Chiral symmetry breaking happen when a chemical and physical processes who do not have any priority in production of one or the other of the enantiomer spontaneously give rise to a large number of one of the enantiomers: left-handed (L) or right-handed (D). Considering the energy, there is equal probability for creating and existence both enantiomers. Chiral asymmetry is found everywhere in nature, and one example are living organisms who utilize L-amino acids and D-sugars. The fact that living organisms choose only one type of amino acids and only one type of sugars as a constituent component has intrigued many scientists. Mechanisms which lead to total chiral purity or to transition from one enantiomer to another are of the great interest.
Sodium chlorate (NaClO3) molecules are achiral but they form chiral crystals in the cubic space group P213.[1] Chiral crystals of sodium chlorate are optically active, and L or D crystals can be easily identified with a pair of polarizers. When polarizers are crossed crystal color changes with rotation of the analyzer (Fig. 1a), i.e. the color of the crystal will be different for clockwise or counterclockwise analyzer rotation depending on the handedness of the crystal (Fig 1b).
Kondepudi et al.[2] showed that stirring during the crystallization of NaClO3 from aqueous solution was sufficient to produce more than 99% of crystals of the same handedness, while in unstirred solution equal number of L and D crystals was obtained. Viedma[3] found that
© Institute of Physics, 2016
Physica Macedonica, Vol. 62
adding glass flasks and stirring the solution which is consisted of the same number of L and D sodium chlorate crystals, leads to complete deracemization of NaClO3, i.e. one enantiomer disappears completely. Chiral symmetry breaking was also reported in experiments with solution temperature changes (heating and cooling cycles)[4] and for other substances [5,6].
Figure 1. a) Crystal color changes with analyzer rotation b) L and D sodium chlorate crystals in solution.
These results are experimental evidence of chiral symmetry breaking on macroscopic scale induced by autocatalysis and tendency for domination of one enantiomer group. One of explanations of the process of chiral symmetry breaking is secondary nucleation. Namely, it is considered that nucleation of chiral ″mother″ crystal influence the production of generations of secundary nucleated crystals of the same handedness.[7-17] In the paper[18] it was shown that chiral symmetry breaking occurs in primary nucleation, and that secondary nucleation only increase it. Due to that, process of chiral symmetry breaking cannot be explained by the nucleation of ″mother″ crystal, i.e. mechanisms which induce domination of one enantiomer group are still unclear.
In order to explain it, different models were suggested and most of them take in consider first phases of the crystallization. Niinomi et al.[19] showed that in crystallization process driven by evaporation of solvent, achiral metastable crystals of sodium chlorate formed prior to chiral crystals in solution growth, and that transition from achiral to chiral state occurred, which means that the chirality of the NaClO3 crystal emerges during the transition. Differences in solubility of metastable achiral and chiral NaClO3 crystals are also important for describing the phenomenon. [20]
2. EXPERIMENTAL PROCEDURE
All experiments were performed with a saturated solution prepared using the following procedure. An aqueous solution of NaClO3 was prepared by dissolving NaClO3 (Analytical grade of substance was 99% purity) in distilled water at room temperature. Then the solution was heated under stirring with a magnetic hot plate stirrer and then left for a three days at saturation
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Biljana Radiša, Mio Mitrovi, Branislava Misailovi: The effect of solution pre-historyon chirality of sodium chlorate crystals from aqueous solutions
temperature to precipitate the excess solute and thus to equilibrate the solution. Crystals were nucleated in the cell by introducing air bubbles through a needle into the cell.
We observed in situ crystal growth by digital optical microscope Nikon SMZ800 equipped with a video recording system (camera Luminera, Infinity 1), using a polarized light. Between 15 and 33 crystal nuclei, sufficiently distant to avoid intergrowth during the growth, were preselected for growth rate measurements during each growth run. In order to investigate the influence of solution history on sodium chlorate crystals growth rate and chirality two types of experiments were performed. In all experiments the solution was saturated at temperature
C)1.00.31(0 °±=T , while the crystal growth was performed at C)1.00.28(r °±=T . Both types of experiments consisted of two parts. In the first part solution temperature
was increased to Tinc=35.0°C (the first type of experiments) or to Tinc = 32.8°C (the second type of experiments), and after that slowly decreased to the growth temperature C)1.00.28(r °±=T . The temperature decreasing in the first type of experiments was about 2.5 h, while in the second type was about 4h. Procedure of both types of experiments was similar and the only difference was in initial and the final intervals of temperature change, showed in Table 1. Temperature was changed from 1T to 2T in steps of T every 5 min. From C8.30 ° to C0.28 ° the solution was left to be cooled by itself, in Type 1 experiments, and also for Type 2 experiments in temperature range from C6.29 ° to C0.28 ° .
Table 1. Temperature changes
Type 1 experiment Type 2 experiment C][ 1 °T C][ 2 °T ]C[°T C][ 1 °T C][ 2 °T ]C[°T 0.35 0.34 5.0 8.32 2.32 3.0 0.34 8.32 4.0 2.32 6.31 2.0 8.32 2.32 3.0 6.31 4.31 1.0 2.32 6.31 2.0 4.31 6.29 05.0 6.31 4.31 1.0 - - - 4.31 8.30 05.0 - - -
Procedure of the second part of both types of experiments was the same. After the temperature of growth was reached, crystals were nucleated by introducing air bubbles through a needle into the cell. Crystal growth lasted about four hours and in all experiments crystals sizes were measured every 40 minutes.
3. RESULTS AND DISCUSSION
At different experimental conditions growth rate dispersions occurred. Histograms representing these dispersions are presented in Figure 2 and Figure 3. These dispersions are described by simple normal distribution, also included in Figure 2 and 3. Zero growth rates, whose existence is completely unclear,21 are excluded from fitting procedure. Several high growth rates at the end of dispersions, which probably pertain to higher activity of dominant dislocation group, are excluded from fitting procedure, too.22
3
Number of measured growth rates N, number of measured growth rates of left-handed crystals NL and number of measured growth rates of right-handed crystals ND, as well as the most probable growth rate R1, R2, R3 and R4 for different experiments are presented in Table 2.
It can be seen from Figures 2 and 3, and Table 2 that crystals grown in Type 2 experiments had higher growth rates. This is in accordance with lattice strain theory.23 Namely, the temperature changes were performed in smaller steps, so crystals had less defects of structure, and consequently higher growth rates.
Table 2. Experimental results
Type 1
N=340 22±4 47.5±0.7 69±2 90±2 NL=155 26±4 51.3±0.5 ND=185 21±3 42.8±0.3 65±3 93±4
Type 2
N=160 25±4 51.8±0.7 83±3 NL=18 47.9±0.5
ND=142 25±1 53.0±0.4 82.8±0.8
Theoretical analysis shows that there is equal probability for creating and existence both enantiomers. Results presented in Table 2 show that in both types of experiments left- and right- handed crystals existed. In Type 1 experiments numbers of L and D crystals are comparable, while in Type 2 experiments right-handed crystals dominated. Due to the fact that in Type 2 experiments the number of L crystals is significantly less than the number of D crystals, conditions might lead to total chiral purity. Also, there was no transition occurred from one enantiomer to another enantiomer.
Regardless this, the influence of crystal growth history is obvious. Further investigations in this field might bring clearer results, and contribute to crystal growth theories development.
Figure 2. Histograms representing growth rate in direction <100> for Type 1 experiments of: a) all observed crystals b) L crystals c) D crystals.
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Biljana Radiša, Mio Mitrovi, Branislava Misailovi: The effect of solution pre-historyon chirality of sodium chlorate crystals from aqueous solutions
Figure 3. Histograms representing growth rate in direction <100> for Type 2 experiments of: a) all observed crystals b) L crystals c) D crystals.
4. CONCLUSIONS
Mechanisms which lead to total chiral purity are of the great interest. In this paper results of investigations of effect of growth history on sodium chlorate crystals chirality are presented. Two types of experiments were performed. Obtained results show that at different experimental conditions growth rate dispersions is occurred. In experiments Type 2 crystals grown at higher rates. In both types of experiments L and D crystals existed. Solution pre-history induced the increase in the population of D-crystals in experiments Type 2. In experiments performed there was no transition from one enantiomer to another. Obtained results indicate that solution pre- history might affect chiral symmetry breaking in crystallization of sodium chlorate crystals even achieving crystals of chiral purity, which request further investigations.
REFERENCES
[1] W.H.Z.Zacharlasen, Kristallograhie 71, 517–529 (1929) [2] D. K. Kondepudi, R. J. Kaufman and N. Singh, Science, 250, 975–976 (1990) [3] C. Viedma, Phys. Rev. Lett., 94, 065504 (2005) [4] K. Suwannasang, A. E. Flood, C. Rougeot and G. Coquerel, Cryst. Growth Des. 13, 3498–
3504 (2013) [5] D. G. Blackmond, Proc. Natl. Acad. Sci. U. S. A., 101, 5732–5736 (2004) [6] J. E. Hein, B. H. Cao, C. Viedma, R. M. Kellogg and D. G.Blackmond, J. Am. Chem. Soc.,
134, 12629–12636 (2012) [7] J. M. McBride, R. L. Carter, Angew. Chem., Int. Ed. Engl. 30, 293−295 (1991) [8] D. K. Kondepudi, K. L. Bullock, J. A. Digits, J. K. Hall, J. M. Miller, J. Am. Chem. Soc.
115, 10211−10216 (1993) [9] R. Y. Qian, G. D. Botsaris, Chem. Eng. Sci. 52, 3429−3440 (1997) [10] R. Y. Qian, G. D. Botsaris, Chem. Eng. Sci. 53, 1745−1756 (1998)
5
[11] T. Buhse, D. Durand, D. Kondepudi, J. Laudadio, S. Spilker, Phys. Rev. Lett. 84, 4405−4408 (2000)
[12] R. Y. Qian, G. D. Botsaris, Chem. Eng. Sci. 59, 2841−2852 (2004) [13] D. K. Kondepudi, C. Sabanayagam, Chem. Phys. Lett. 217, 364−368 (1994) [14] B. Martin, A. Tharrington, X.-I. Wu, Phys. Rev. Lett. 77, 2826−2829 (1996) [15] J. H. E. Cartwright, J. M. García-Ruiz, O. Piro, C. I. S. Díaz, Tuval, Phys. Rev. Lett. 93,
035502 (2004) [16] J. H. E. Cartwright, O. Piro, Tuval, I. Phys. Rev. Lett. 98, 165501 (2007) [17] M. Uwaha, J. Phys. Soc. Jpn. 73, 2601−2603 (2004) [18] C. Viedma, J. Cryst. Growth 261, 118−121 (2004) [19] H. Niinomi, T. Yamazaki, S. Harada, T. Ujihara, H. Miura,Y. Kimura, T. Kuribayashi, M.
Uwaha, and K. Tsukamoto, Cryst. Growth Des. 13, 5188−5192 (2013) [20] H. Niinomi, A. Horio, S. Harada, T. Ujihara, H. Miura, Y. Kimura, K. Tsukamoto, J. Cryst.
Growth 394, 106–111 (2014) [21] B. M. Misailovi, D. A. Malivuk, A. A. eki, M. M. Mitrovi, Cryst. Growth Des. 14,
972-978 (2014) [22] M. M. Mitrovi, A. A. eki, B. M. Misailovi, B. Z. Radiša, Ind. Eng. Chem. Res. 53
(50), 19643-19648 (2014) [23] A. E. D. M. van der Heijden, J. P. van der Eerden, J. Cryst. Growth 118, 14-26 (1992)
6
NORMALIZED INTEGRAL FOURIER TRANSFORM AMPLITUDE FOR FOOD QUALITY DETERMINATION BY
ACOUSTIC IMPULSE RESPONSE METHOD
1Faculty of Engineering and Technology, Department of Mathematics and Physics, Higher Colleges of Technology, Ras al Khaimah Men’s Campus, United Arab Emirates
2 Faculty of Natural Sciences and Mathematics, State University of Tetovo, bul. Ilinden, b.b., 1200 Tetovo, Republic of Macedonia
Abstract. The research has been conducted with the goal of determining the shelf-life of post harvested tomatoes by implementing the non-destructive physical method of acoustic impulse response. Normalized integral Fourier transform amplitude has been introduced for enhancing the precision of shelf- life determination in comparison to the use of the well-known firmness parameter Fi. The firmness coefficient Fi has been calculated on the basis of the dominant (fundamental) frequency of tomato response signal f and sample mass m. The Fast-Fourier transform algorithm has been used on the acoustical time dependent electrical signals taken from an acoustical sensor attached to the product under investigation, thus allowing the frequency dependent Fourier transform to be found.
PACS: 43.40.+s, 45.10.-b, 46.40.-f, 87.19.r-
1. INTRODUCTION
For assessing the quality of agricultural products beyond the use of chemical and biochemical methods, great practical importance has been given recently to the physical methods, especially the non-destructive methods that are fast, accurate and reliable [1.2]. One such method is the acoustic impulse response method, which has been investigated by many researchers aiming at measuring the firmness of fruits and vegetables [3-5]. The method gives solid results that were proven by the classical destructive firm testing technique [6]. The main advantage of the acoustic impulse response method is that it is fast, reliable, non-destructive and can be performed in-field (on-site). The main disadvantage of the method is that it gives a “bulk” information for the product, which means the acoustic response is like averaged for the whole sample. Only the magnetic resonance imaging method (MRI) can give perfect picture of the inside structure of the food product examined, but this method requires expensive equipment and highly trained examiner [7]. The major achievement of the acoustic impulse response method is seen in the context of good correlation of the results for maturity stage of food samples with the results from
many other methods [8]. That means that the acoustic impulse response method can be used by alone.
2. MATERIALS AND METHODS
The “Melody” variety tomatoes (Lycopersicum) were hand-harvested from the greenhouse orangery of the company “Agroproduct” from Gostivar, Republic of Macedonia, on October 17, 2013. Twenty seven tomatoes were sorted according to their mass (250 g ± 50 g) and color appearance of same class (90% of the surface to be red and 10% of the surface showing change from green to yellow). For that purpose, a color sensor panel have been composed of five assessors experienced in tomato assessment. Tomatoes were harvested from the tomato trees between 40 – 45 cm above the ground. Only one tomato was taken from a tomato tree. The tomatoes were grown according to the following conditions: average temperature of 14 – 16 0C during nights; water supply of 1.5 L for 24 hours; fertilizer “Jara” of 180 g per tomato tree; pesticides treatment every week; insecticides treatment on occasions. Tomatoes were stored at 15 ± 1 0C, relative humidity 60% and were placed in a box that can hold each tomato separated. After been harvested, tomatoes were transferred to the lab and kept during the investigation on average temperature of 15 ± 1 0C and relative humidity 60 % on shelves with artificial illumination same as under selling condition on the local market.
According to the acoustic impulse response method, tomatoes were impacted without a damage with a small ball pendulum and tomatoes vibration response signals were recorded with a microphone as a sensor. The firmness coefficient Fi has been calculated on the basis of the dominant (fundamental) frequency of tomato response signal f and sample mass m. The firmness coefficient Fi, which for spherical-like fruits was first proposed by Abbott [9] and was modified by Cooke and Rand [10], has been calculated according to the following formula:
2 2 3
iF f m= . (1)
Same method has been applied from many researchers on various products (apples, watermelons, mangos, and many others) [11, 12]. To reduce errors due to sample internal homogeneity differences as well as injuries after the collision with the impactor (pendulum ball), it is preferred to change the position of impact and the average response is obtained after three hits on the equatorial surface of the samples when the microphone is positioned at angle 1200 in respect to the impact [10].
The dominant or fundamental frequency of vibration of the food product has been determined by doing Fourier transform of the amplitude-time function of the sensor signal which gave the amplitude-frequency function (so called power spectrum). There are several methods described in the literature for calculating the Fourier transform and we used the Fast Fourier Transform algorithm incorporated in the powerful OriginPro® graphical analysis software. This software combines some good features from MatLab® and Mathematica®.
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S. Rendevski1, I. Aliu and N. Mahmudi: Normalized integral fourier transform amplitude for food quality determination by acoustic impulse response method
Figure 1. Typical acoustic signal. Figure 2. Fourier transform of the signal from fig.1
On figure 1, a common acoustical sensor signal is given after impact on a tomato, and on figure 2, the Fourier transform is given for the same function. It is obvious that the dominant frequency f of the sample vibration has a value of 95 Hz, but there is also another small peak in the function at 180 Hz and on some cases there were more. The position of the main, or the dominant frequency, and the presence of other peaks in the Fourier transform can give more information about the food product. For example, if there is second smaller peak, that means the vibration of the sample is more complex, probably because of different internal structure inside the product – already rotten regions, over-ripen small regions that will spread in the following days, large inside inhomogeneity in density of the product because of changes in grown conditions, and so on. When focused only of the main peak or the dominant frequency of vibration, the shift over time to lower values indicates softer product and/or separations inside of the product – one larger region with more water content and another (a core) denser (less water content) which is typical for over-mature stage of fruits and vegetables.
Some detailed information from the acoustic impulse response method as well as the use of piezoelectric or microphone as a sensor applied for tomato quality assessment after storage has been given in literature [2-5]. In our research, we used equipment schematically described on figure 3. Here, we have a ring sample holder that gently hold the tomato sample; impactor with a pendulum ball (self-made with aluminum ball of 20 g mass and pendulum angle 300 that catch the ball back after the first impact); microphone model MI-10, A4 Tech, Taiwan, with integrated amplifier and a computer for data acquistion and calculations. GoldWave® audio studio software has been used for taking the amplitude-time signal from the microphone. The sampling rate was 20,000 points per second.
In this paper, a new quantity determined from the Fourier transform of the acoustical signal, named normalized integral amplitude, Nia, has been introduced (in units of Hz). The normalized integral amplitude is obtained after integration of the Fourior transform function on the whole frequency range and by division after by the amplitude at the dominant frequency.
9
Figure 3. Scheme of the equipment for acoustic impulse response measurements.
From the physics of sound and mathemaics of the Fourier transform it is known that according to the Parseval’s theorem the energy of the acoustical signal received by the microphone
2
( )f t dt+∞
−∞∫ is equal to the energy of the vibrational spectrum of the sample obtained from the Fourier transform
2
( )F dω ω+∞
−∞∫ [13, 14]. Knowing this, it is obvious that Nia is directly proportional to the energy of the signal (or the vibrational spectrum) and after normalizing it to the maximum amplitude, Nia can be considered as a parameter that gives information on the “dissipation” of the energy not only around the dominant frequency of vibration, but on the whole spectrum, indeed.
In the literature, when acusto-vibrational properties are expressed with only the dominant frequency, the contribution to all other frequencies of vibration in the obtained Fourier transform function is taken as negligible. The aim here is not to connect Nia quantity to the firmness properties of samples, as Abbott (1968) and Cooke and Rand (1973) did, but to check the behavior of the quantity over storage time passed.
Figure 4. Graphical presentation of the changes of f, Nia and Fi over time.
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S. Rendevski1, I. Aliu and N. Mahmudi: Normalized integral fourier transform amplitude for food quality determination by acoustic impulse response method
It is expectable to have drastic change of Nia value if several more vibration characteristics exist (peaks) beside the dominant one in the sample. Aiming to understand the importance of Nia quantity, the Fourier transform function for over-matured sample will be discussed in comparison to same sample for which the Fourier transform function has been taken immediately after harvesting. The values of Nia parameters were 168 Hz and 202 Hz, respectively, while the values of the dominant frequency where 73 Hz and 102 Hz, respectively. In the research presented it has been found that when the difference between the Nia value and f is more than double, the product can be considered as over-matured, as in the above example when we found f = 73 Hz and Nia = 168 Hz. When time dependences of Nia and f is constructed, there is more drastic change of Nia value than f value after some critical moment (fig.4). This moment is direct indication of the shelf-life of the tomatoes because after this critical time the quality of the sample deteriorates significantly.
3. RESULTS, DISSCUSIONS AND CONCLUSIONS
The mass of the tomatoes are important quantities for determination of the firmness parameter, according to Eq.1. The mass was measured each experimental day using the electronic weighing scale with precision of 1 g. Mass loss was calculated according to the equation
[( ) / ] 100%l i f im m m m= − ⋅ , where ml is the reduced mass in percentage, mi is the initial mass and mf is the final mass of the product, in grams. It has been found during the research that the reduced mass was about 5% of the initial mass of all tomatoes. This is due to the water loss during the 11 days period of investigation. However, such a loss of mass was not considered influential on the change in maturity stage of the product in investigated period.
The values of the dominant (fundamental) frequency of vibration f, normalized integral amplitude Nia, firmness parameter Fi calculated from Eq.1, are given in Table 1.
Table 1. Main parameters obtained from the acoustic impulse response method.
Time (in days) f,Hz Nia,Hz Fi, kg2/3/s2
0 108 ± 11 239 ± 18 4017 4 103 ± 10 210 ± 20 3592 7 84 ± 15 205 ± 22 2400 9 73 ± 17 138 ± 20 1568 11 66 ± 12 142 ± 23 1458
From fig.4, it is obvious that the change of Nia over time are more pronounced in comparison to the changes of f. The change of Nia is following an inverse S graphical dependence from which it is visible that after 9 days of storage there is significant decrease of Nia value which indicates over-mature stage of the tomatoes. The same behavior has been found when changes of the firmness parameter Fi have been followed over days passed (fig.4). This indicates that Nia quantity can be considered as a measure for the vibration characteristics of the tomato, instead of the single dominant frequency of vibration. The dependence of Nia over time is following the dependence of Fi in more profound way than the f. That implies that there is direct connection between Nia and Fi that requires theoretical investigation henceforth.
11
As seen from the results from the application of the acoustic impulse response methods, for the “Melody” type tomatoes kept at 15 0C at 60 % relative humidity and normal illumination, the shelf-life has been found for nine days of storage.
The method of acoustic impulse response provides a “view” of the internal homogeneity of tomatoes without destructing them and for this reason it is considered as a method of non- destructive testing of products. If there is inhomogeneity inside tomatoes, that will be reflected through different dominant frequencies and normalized integral amplitude of the product vibrations during slight impact of the tomatoes in different positions, for example, around the stem, at the bottom of the fruit or in the equatorial part.
REFERENCES
[1] Gunasekaran, S. “Nondestructive food evaluation: Techniques to analyze properties and quality”. New York, USA: CRC Press (2000)
[2] Kilcast, D. “Instrumental assessment of food sensory quality: A practical guide”. Cambridge, UK: Woodhead Publishing (2013)
[3] Galili, N., Shmulevich, I., Benichou, N. Transactions of the American Society of Agricultural Engineering, 41, 399-407 (1998)
[4] Duprat, F., Grotte, M., Pietri, E., Loonis, D., Studman, C.J. Journal of Agricultural Engineering Research, 66(4), 251-259 (1997)
[5] Aboonajmi, M., Jahangiri, M., Hassan-Beygi, S.R. Journal of Food Processing and Preservation, 39(6), 3175-3188 (2015)
[6] Barrett, D.M., Beaulieu, J.C., Shewfelt, R. Critical Reviews in Food Science and Nutrition, 50(5), 369-389 (2010)
[7] Zhang, L., McCarthy, M.J. Postharvest Biology and Technology, 67, 37-43 (2012) [8] Arana, I. “Physical properties of food”. Boca Raton, USA: Taylor & Francis Group (2012) [9] Abbott, J.A., Bachan, A., Childers, G.S., Fizterald, J. V., Matusik, F. J. Food Technology,
22, 635-645 (1968) [10] Cooke, J. R., Rand, R. H. Journal of Agricultural Engineering Research, 18, 141-157
(1973) [11] Abbott, J.A. Quality of Fresh and Processed Fruits, 542, 265-279 (2004) [12] Butz, P., Hofmann, C., Tauscher, B. Journal of Food Science, 70(9), 131-141 (2005) [13] Pain, J.H. “The physics of vibration and waves”. Chichester, UK: John Wiley and Sons
(2005) [14] Urumov, V. “Mathematical physics: methods, applications and problems (in Macedonian)”.
Skopje, Republic of Macedonia: Prosvetno Delo (1997)
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NONLINEAR ULTRASOUND WAVE GENERATION IN QUASI-TWO DIMENSIONAL ORGANIC CONDUCTORS
D. Krstovska and B. Mitreska Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and
Methodius University, P. O. Box 162, 1000 Skopje, Macedonia
Abstract. The amplitude of an ultrasound wave with double frequency 2ω generated by Joule heating as a heat source in the quasi-two dimensional layered organic conductor β−(BEDT-TTF)2IBr2 is analyzed as a function of the magnetic field strength and its orientation with respect to the plane of the layers. Angular oscillations of the ultrasound wave amplitude are correlated with the changes of electrical and thermal conductivity of the compound when the magnetic field is tilted from the normal to the layers plane due to the periodic dependence of the carriers velocity along the less conducting axis on the field orientation. The period of oscillations allows the characteristic parameters of the Fermi surface to be determined. The magnetic field dependence reveals that the effect of nonlinear generation of the ultrasound wave due to Joule heating is quadratic in proportion to the magnetic field at each field direction. The behavior of the wave amplitude is different depending on the field orientation from the normal to the layers, i.e., is it oriented at the maximum or minimum in the angular dependence. In layered organic conductors due to the small electron mean free-path the nonlinear generation of ultrasound waves can be observed for a wide range of magnetic fields providing possibilities for experimental studies using non-contact ultrasonic techniques.
PACS: 74.70.Kn, 72.15.Jf, 72.50.+b
1. INTRODUCTION
The mechanisms of linear transformation, which are responsible for the generation of ultrasound in conducting media at the frequency of an electromagnetic wave ω incident on the surface of the metal, are a standard subject of investigation in electromagnetic-acoustic conversion problems [1-6]. In conducting media, the mechanism occurs as follows: when an electromagnetic wave with frequency ω is incident on the conductor, nonuniform temperature oscillations of the same frequency appear as a result of the thermoelectric effect. These oscillations, in turn, generate acoustic oscillations in the conductor with a frequency that coincides with the frequency of the incident electromagnetic wave (contactless excitation of ultrasound).
The induction [7-10] and deformation force [11-18] have been studied in greatest detail as sources of linear transformation of waves. In the last few years the focus of investigations of electromagnetic excitation of ultrasound has shifted to the study of subtle nonlinear conversion mechanisms. Some studies in ordinary metals have shown that these mechanisms could also generate ultrasound wave with frequency 2ω, i.e., in the nonlinear regime [19-23].
Apart from the induction and deformation forces, longitudinal ultrasound at double frequency can also be generated by other source of nonlinearity. The most important such source is the force arising due to the appearance of thermoelastic stresses in the Joule-heated skin layer [24]. Joule heating is the characteristic of material that can be heated up when there is current flow because electrical energy is transformed into thermal energy. The thermoelectric mechanism of ultrasound generation is in principle a nonlinear interaction due to the action of the thermoelastic stresses on the crystal lattice. These stresses that are caused by the temperature oscillations with frequency 2ω (Θ∼cos(2ωt)) arising from the alternating part of the Joule heat generate ultrasonic waves with double frequency.
In this paper we study the effect of Joule heating from electric currents flowing through a quasi-two dimensional organic conductor on the temperature distribution and nonlinear generation of a longitudinal ultrasound within the conductor. The quasi-two dimensional (q2D) charge-transfer salts form the largest organic superconductor family with unusual normal and superconducting properties which come out of the layered structure as well as the specific shape of the Fermi surface. Their most remarkable feature is the reduced dimensionality of the electronic band structure caused by the specific character of their crystal structure. Specifically, the q2D organic conductor β−(BEDT-TTF)2IBr2 is considered and the amplitude of the generated longitudinal ultrasound is analyzed as a function of the magnetic field strength and its orientation from the less conducting axis to the plane of the layers.
2. FORMULATION OF THE PROBLEM
The ultrasound with double frequency is generated as a result of temperature oscillations which are induced by Joule heating as a heat source. Here we consider a case when electric currents flowing through the conductor are along the most conducting axis (x-axis), j = (jx, 0, 0) and the only nonzero component of the electric field is the x−component, E = (Ex, 0, 0). An ultrasound of double frequency is generated and propagating along the less conducting axis (z- axis), k = (0, 0, k), and therefore all of the quantities depend only on the z−component. The conductor is placed in magnetic field oriented at an angle θ from the normal to layers plane, in the yz plane, B = (0, Bsinθ, Bcosθ). A temperature oscillating with frequency 2ω can occur if 2ωτ<<1 where τ is the relaxation time of the conduction electrons. We study the case of normal skin effect when the condition ,1<<lkT where l is the electron mean-free path length and kT is the thermal wave number, is satisfied.
The complete system of partial differential equations, describing the generation of longitudinal ultrasound at frequency 2ω, includes Maxwell's equations for the magnetic B and electric E field, the kinetic equation for the nonequilibrium correction χ to the electron distribution function, and the equations of heat conduction and elasticity:
[ ]0
z z z ω
14
D. Krstovska and B. Mitreska: Nonlinear ultrasound wave generation in quasi-two dimensional organic conductor
Here v and p are the electron velocity and momentum, e is the electron charge, Θ is the high-frequency addition to the mean temperature T of the crystal, QJ is the power density of the heat source, C is the volumetric heat capacity, zzκ is the thermal conductivity tensor, ks = 2ω/s is the wave vector of the ultrasonic wave at frequency 2ω, s is the ultrasound wave velocity and β is the volumetric expansion coefficient.
The above system of equations must be supplemented with the corresponding boundary conditions for the heat flux and the ultrasound amplitude at the conductor’s surface. In the case when the heat source is the time-dependent part of the Joule heating the boundary conditions are:
2 0 0 0| 0, | | .zz z z z
U z z z
∂ ∂ ∂ (2)
By using Maxwell’s equations one obtains the following expression for the Joule heating
2
2 ( )
0
µ −= = =jE , (3)
where (1 )k i δ= + and 02 ( ).xxδ ωµ σ= Substituting eq. (3) into the heat conduction equation (third equation in (1)) we obtain the
following expression for the temperature distribution
22
zz
∂ (4)
where (1 )T Tk i δ= + and .T zz Cδ κ ω= Here Tδ is the penetration depth of the thermal field under the conditions of a normal skin effect.
The solution of eq. (4) is given as
2
− Θ =
− (5)
The complex amplitude of the generated longitudinal ultrasound wave with double frequency can be written in the following form
2 2
= +
(6)
The components of the conductivity tensor which relate the current density to the electric field can be calculated by using the Boltzmann transport equation for the charge carrier distribution function, based on the tight binding approximation band structure within the single relaxation time approximation τ (second equation in (1)) [25]. The components of the electrical conductivity tensor are determined as follows
15
∂ =
∂∫ ∫ ∫
(7)
Here is the Planck’s constant divided by 2π, pB =pysinθ +pzcosθ =const is the momentum projection in the magnetic field direction, and t is the time motion of the conduction electrons in the magnetic field under the influence of the Lorentz force dp/dt =e (v×B).
For the quasi-two dimensional organic conductor whose electron energy spectrum has the form
2 2
ε η +
, (8)
where m* is the electron cyclotron effective mass, η is the quasi-two dimensionality parameter, vF is the characteristic Fermi velocity of the electrons along the layers, c is the lattice constant, the expressions for zzσ and xxσ take the following form
2
tan cos ( ) and ( tan ).
θ = = +
(9)
Here 0σ coincides in order of magnitude with the electrical conductivity along the layers in the absence of a magnetic field, Dp = 2pF is the averaged diameter of the Fermi surface along the py axis and h = e/m∗ τ.
The thermal conductivity zzκ is obtained by using the Wiedemann-Franz law
2 2
3. RESULTS AND DISCUSSION
The amplitude of the generated ultrasound wave with double frequency due to Joule heating is a function of the frequency of applied electric current ω, the magnetic field B, the angle between the normal to the layers and the magnetic field θ, as well as of the characteristics of the q2D conductor (electrical and thermal conductivity). For the q2D organic conductor β−(BEDT- TTF)2IBr2, the interlayer transfer integral and the cyclotron mass extracted from data on magnetic quantum oscillations are tc = 0.35 meV and m* = 4.2me, respectively and the relaxation time is of the order of 10 ps [26]. The Fermi surface is a weakly warped cylinder and the quasi-two dimensionality parameter is of order η = 10−3.
Figure 1 shows the angular dependence of the amplitude U2ω(θ) for several field values with pronounced peaks at certain angles whose height decreases with increasing angle between the normal and the plane of the layers. There is a deep minimum for the orientation of the magnetic field in the plane of the layers. The angular oscillations are characteristic of the kinetic and
16
thermoelectric coefficients of layered organic conductors. The oscillatory dependence of the ultrasound wave amplitude U2ω on the angle θ is determined by the angular oscillations of the magnetoresistance 1−= zzzz σρ as well as thermal conductivity zzκ which are associated with the charge carriers motion on the cylindrical Fermi surface in a tilted magnetic field.
Figure. 1: Angular dependence of the ultrasound wave amplitude U2ω for T = 30 K, η = 10−3 and several field values B = 3, 4.5, 6, 8, 10 and 12 T from bottom to top.
In the vicinity of angles where the amplitude U2ω has maximum value, ,max nn θθ = the
average drift velocity zv of charge carriers along the ultrasound wave vector coincides with the velocity s of the ultrasound wave, and their interaction with the wave is most effective. As a result, at these angles the amplitude is the largest (especially this trend is apparent for θ > 45) and U2ω
(θ) exhibits peaks in the angular dependence. The magnitude of the peaks is modulated with changing the tilt angle θ due to the oscillating zeroth order Bessel function )./tan(0 θpcDJ
If the vectors ks and B are not orthogonal, the average velocity of charge carriers in the direction of propagation of the wave differs from zero for any shape of the Fermi surface, i.e., charge carriers drift in the direction of wave propagation. The existence of points at which the interaction with the wave is most effective on different trajectories leads to a resonant dependence of the acoustic wave amplitude on magnetic field B and the angle between the normal to the layers and magnetic field θ. At θ < π/2 the charge carriers which are in phase with the acoustic wave propagating along the z−axis are the carriers with time-averaged drift velocity Fz vv η= in the direction of propagation of the ultrasound wave.
The interaction of these electrons with the ultrasound wave is most effective when 1<<Tsk δ under the condition of normal skin effect, ,1<<lkT and when the q2D closed orbits are
not strongly warped. For θ < π/2 all orbits of electrons whose states belong to the corrugated cylinder are closed and do not contain self-intersections. Cross-sections of the corrugated cylinder with the plane pB = const become more elongated with tilting the field away from the z−axis. For θ close to π/2 there is a significant transformation of the electron trajectories in the momentum space. Due to the pronounced elongation of the closed orbits the amplitude of the ultrasound wave U2ω rapidly decreases with increasing angle as seen in Fig. 1.
17
For the amplitude U2ω plotted versus tanθ (Fig. 2), the same trends are apparent, but important feature to emphasize is that the amplitude U2ω is nonzero in the whole range of angles in comparison to the amplitude of the fundamental wave Uω generated by the thermoelectric forces in the linear electromagnetic-acoustic transformation processes in q2D organic conductors [27]. This is because of the sinθ factor in the denominator of the expression for the amplitude U2ω (eq. 6).
Figure. 2: tan θ dependence of the ultrasound wave amplitude U2ω for T = 30 K, η = 10−3 and several field values B = 3, 6, 8, 10, 12, 15 and 20 T from bottom to top. The inset shows the tan θ dependence of the amplitude U2ω for T = 30 K, η = 10−3 and B = 20 T with the corresponding Yamaji angles where the amplitude U2ω has maximum or minimum value.
When /tanθpcD equals a zero of the zeroth-order Bessel function, then at that angle the electrical conductivity along the less conducting axis, ),(θσ zz will be negligible and the magneto-resistance )(θρ zz will be a maximum. These angles are known as Yamaji angles [28, 29]. If 1/tan >>θpcD , then the zeros occurs at angles ,max
nn θθ = given by /tan max npcD θ =
π(n −1/ 4), n = 0, 1, 2, 3, … For )(θρ zz to be a minimum it should be ,min nn θθ = where
/tan min npcD θ = π(n +1/ 4). According to eqs. (6) and (9), the amplitude U2ω has a peak or dip at
angles where )(θρ zz is maximum or minimum as expected in case when there exist only closed Fermi surface sections. This is plotted in the inset of Fig. 2 for B = 20 T where the corresponding angles are marked at which the amplitude U2ω has a peak or dip, respectively. Also, the tan θ position of the peaks and dips is field independent, i.e., does not change with increasing field. The narrow peaks that appear in the U2ω (tan θ) dependence repeat with a period ./2)(tan pcDπθ = The experimental measurements of the period will allow the diameter Dp of the Fermi surface to be determined. In addition, from the shift between the tan θ positions where U2ω (tan θ) is minimum and tan θ positions where U2ω (tan θ) has a peak the value of the interlayer transfer integrals tc in the energy dispersion could be calculated.
Next we look at the curves of the amplitude U2ω as a function of magnetic field at fixed angles. Particularly, the angles where U2ω (θ) has maximum and minimum value, determined from the inset of Fig. 2, will be considered. Figure 3 shows the traces of U2ω (B) for several field
18
directions from the normal to the layers: θ = 3, 10, 15, 28, 50.05, 64.78, 75, 83 and 90. The effect of generation of the ultrasound wave due to Joule heating is quadratic in proportion to the magnetic field at each field direction. It is important to note that the field behavior of the ultrasound wave amplitude U2ω is quite distinct for max
nn θθ = and .min nn θθ = While for angles
where the amplitude U2ω has a peak in the angular dependence, for example at the firs peak 05.50max
1 =θ in Fig. 3, it is increasing with increasing field without any changes, for angles where the amplitude U2ω has a dip, as for ,78.64min
1 =θ there is a narrow peak in the magnetic field
dependence at lower fields (inset of Fig. 3). This peak is also present for small angles from the z- axis but with smaller amplitude compared to that for min
nn θθ = as shown in the inset of Fig. 3.
Figure. 3: Magnetic field dependence of the ultrasound wave amplitude U2ω for T = 30 K and several field directions from the normal to the layers: θ = 3, 10, 15, 28, 50.05, 64.78, 75, 83 and 90. The inset shows the low field behavior of U2ω for T = 20 K, η = 10−3 and θ = 3, 10, 15, 28, 64.78 and 90.
This implies that at angles where the amplitude U2ω is small, below 45=θ (tan θ < 1) and at ,min
nn θθ = compared to its magnitude at the peaks there is always a narrow peak in the field dependence. The appearance of the peak could be associated with a resonant condition in the dependence of the ultrasound wave amplitude on magnetic field B due to the large local temperature gradient induced by the Joule heating at lower fields. The position of the peak and the fulfillment of the resonant condition are field and angle dependent.
The observation of nonlinear ultrasound wave generation due to Joule heating as a heat source is constraint by the condition 2ωτ<<1. In the organic conductor β−(BEDT-TTF)2IBr2 this condition is always fulfilled, even at high frequencies ω = 108-109 Hz, as the relaxation time τ is very small (for β−(BEDT-TTF)2IBr2, τ = 10 ps [26]). In addition, for the generation of waves to be the most effective other conditions must be satisfied. This includes fulfillment of 1<<Tsk δ as long as the conditions for normal skin effect Tl δδ ,<< are satisfied. The Fermi velocity of β−(BEDT-TTF)2IBr2 is vF = 1.5⋅105 m/s [30] and the electron mean free-path is of order l = vF τ = 1.5 µm. The skin depth of the β-(BEDT-TTF)2IBr2 crystals is about δ = 0.12 mm [30]. It is evident that the necessary conditions are fulfilled even at higher fields providing the ultrasound wave generation with double frequency to be studied experimentally in a wide range of magnetic fields. Experiments on thermoelectric generation of ultrasound waves in layered organic conductors
19
would be highly favorable since these compounds are synthesized with high purity that makes them very convenient for performing experiments.
4. CONCLUSIONS
The nonlinear generation of an ultrasound wave of double frequency (ω = 108-109 Hz) in layered quasi-two dimensional organic conductors as a result of Joule heating as a heat source is considered. The amplitude of the generated wave due to thermoelectric stresses caused by the alternating part of the Joule heat is analyzed as a function of the magnetic field B, the angle between the normal to the layers and the magnetic field θ as well as of the characteristics of the conductor. Specifically, the parameters for the organic conductor β−(BEDT-TTF)2IBr2 are used to obtain the ultrasound wave amplitude U2ω at temperature T = 30 K and the quasi-two dimensionality parameter η = 10−3, i.e., for not strongly warped Fermi surface. It has been shown that the necessary conditions for observing the nonlinear generation of ultrasound wave with double frequency are always fulfilled even in relatively high but nonquantizing magnetic fields.
The oscillatory dependence of the ultrasound wave amplitude U2ω on the angle θ is determined by the angular oscillations of the magnetoresistance )(θρ zz which are associated with the charge carriers motion on the cylindrical Fermi surface in a tilted magnetic field. At angles where the amplitude U2ω has maximum value, the average drift velocity zv of charge carriers along the ultrasound wave vector coincides with the velocity s of the ultrasound wave and their interaction with the wave is most effective. Therefore narrow peaks appear in the angular dependence of the amplitude U2ω that correspond to the most effective interaction of the charge carriers with the wave. From the period of the angular oscillations the diameter of the Fermi surface as well as the interlayer transfer integrals in the energy dispersion can be determined.
The magnetic field dependence of the amplitude U2ω reveals existence of peaks present for small angles from the z-axis as well as for angles where the amplitude U2ω has a dip in the angular dependence. The appearance of the peak in the field dependence could be associated with the resonant dependence of the ultrasound wave amplitude on magnetic field B at lower fields.
REFERENCES
[1] W. D. Wallace, Int. J. NDT 2, 309 (1971) [2] E. R. Dobbs, p. 127-150, in “Physical Acoustics. Principles and Methods”, Edited by W. P.
Mason, Volume 10, Academic, New York (1973) [3] H. M. Frost, pp. 179-275, in “Physical Acoustics”, Edited by W. P. Mason & R.N. Thurston,
Volume 14, Academic Press, New York (1979) [4] G. A. Budenkov and S. Yu. Gurevich, Defektroskopiya 5, 5 (1981) [5] A. N. Vasil'ev and Yu. P. Gaidukov, Usp. Fiz. Nauk 141,431 (1983) [Sov. Phys. Usp. 26 (1
l), 952 (1983)] [6] E. Kartheuser, R. L. Ram Mohan and S. Rodriguez, Adv. Phys. 35,423 (1986) [7] V. M. Kontorovich and A. M. Glushyuk, Zh. Eksp. Teor. Fiz. 41, 1195 (1961) [Sov. Phys.
JETP 14(4), 852 (1962)]
20
[8] V. Ya. Kravchenko, Zh. Eksp. Teor. Fiz. 54, 1494 (1968) [Sov. Phys. JETP 27(5), 801 (1968)]
[9] V. L. Fal'ko, Zh. Eksp. Teor. Fiz. 85, 300 (1983) [Sov. Phys. JETP 58(1), 175 (1983)] [10] A. N. Vasil'ev, Yu. P. Gaidukov, M. I. Kaganov et al., Fiz. Nizk. Temp. 15, 160 ( 1989)
[Sov. J. Low Temp. Phys. 15, 91 (1989)] [11] G. Turner, R. L. Thomas and D. Hsu, Phys. Rev. B 3, 3097 (1971) [12] W. D. Wallace, M. R. Gaerttner and B. W. Maxfield, Phys. Rev. Lett. 27, 995 (1971) [13] D. E. Chimenti, C. A. Kukkonen and B. W. Maxfield, Phys. Rev. B 10, 3228 (1974) [14] D. E. Chimenti, Phys. Rev. B 13, 4245 (1976) [15] N. C. Banic and A. W. Overhauser, Phys. Rev. B 16, 3379 (1977) [16] E. A. Kaner, V. L. Fal'ko and L. P. Sal'nikova, Fiz. Nizk. Temp. 12, 831 (1986) [Sov. J.
Low Temp. Phys. 12, 471 (1986)] [17] E. Kartheuser and S. Rodriguez, Phys. Rev. B 33, 772 (1986) [18] M. A. Gulvanskii, Fiz. Tverd. Tela 29, 1249 ( 1987) [Sov. Phys. Solid State 29, 715
(1987)] [19] Yu. I. Sazonov and Yu. M. Shkarlet, Defektoskopiya, No. 5, 1 (1969) [20] I. Cap, Acta Phys. Slov. 32, 77 (1982) [21] A. P. Korolyuk and V. I. Khizhnyi, Pis'ma Zh. Eksp. Teor. Fiz. 48, 348 (1988) [JETP Lett.
48, 385 (1988)] [22] A. N. Vasil'ev, M. A. Gulyanskii and M. I. Kaganov, Zh. Eksp. Teor. Fiz. 91, 202 (1986)
[Sov. Phys. JETP 64(1), 117 (1986)] [23] N. M. Makarov, F. P. Rodriguez and V. A. Yampol'skii, Zh. Eksp. Teor. Fiz. 95, 205
(1988) [Sov. Phys. JETP 67(9), 1943 (1988)] [24] L. D. Favro, IEEE Ultrason. Symp. Proc. 1, 399 (1986) [25] A. A. Abrikosov, pp. 201-202, in “Fundamentals of the Theory of Metals”, Volume 2,
North-Holland, Amsterdam (1988) [26] M. V. Kartsovnik, Chem. Rev. 104, 5737 (2004) [27] D. Krstovska, O. Galbova and T. Sandev, EPL 81 37006 (2008) [28] K. Yamaji, J. Phys. Soc. Jpn. 58 1520 (1989) [29] R. Yagi, Y. Iye, T. Osada and S. Kagoshima, J. Phys. Soc. Jpn. 59 3069 (1990) [30] A. Chernenkaya, A. Dmitriev, M. Kirman, O. Koplak and R. Morgunov, Solid State
Phenomena 190, 615 (2012)
ON TWO EXTREMELY FAST METHODS FOR THE SOLUTION OF THE NON-LTE LINE FORMATION PROBLEM
O. Atanackovi Department of Astronomy, Faculty of Mathematics, University of Belgrade,
Studentski trg 16, Belgrade, Serbia
Abstract. The solution of radiative transfer (RT) is a necessary step in the complex process of an astrophysical object modeling. In order to properly determine the physical and dynamical properties of an object from its spectrum, a reliable numerical method that can handle the multilevel non-LTE radiative transfer in spectral lines is needed. This problem is difficult to solve due to the non-local and strongly non-linear coupling between the atomic level populations (i.e. excitation state of the gas) and the radiation field intensities in the corresponding line transitions. For its solution an iterative method is required. For the solution of more complex, time-dependent or multi- dimensional RT problems, iterative method should be as fast as possible. Here we shall describe two methods, Iteration Factors Method (IFM) and Forth-and- Back Implicit Lambda Iteration (FBILI) that, owing to the use of the so called iteration factors in an explicit and implicit way respectively, converge extremely fast to the exact solution.
Key words: Radiative transfer, Numerical methods, Stellar atmospheres
1. INTRODUCTION
The spectra of astrophysical objects contain a wealth of information on their chemical composition and physical properties. The principal goal is to develop the methods to extract this information from their spectra. In practice, the so-called synthetic approach is applied: starting from the postulated model of an object and the theory of radiative transfer (RT) one computes the emergent spectrum and compares it to the observed one, produced by the real object and its own energy transport mechanisms. The model and the theory of radiation transport are to be modified until the agreement between the computed and the observed spectrum is reached. Therefore, the solution of radiative transfer in a given astrophysical object is an inevitable step in the object modeling.
The spectral lines are the best diagnostic tool for interpreting the physical properties of celestial bodies. Hence, the understanding of line formation, i.e. radiative transfer in spectral lines is of fundamental importance and deserves a special attention. We shall focus here on stellar spectra, since the stellar atmospheres are the most studied and the best understood astrophysical objects. However, the methods we consider here can be easily generalized to the study of RT in many other media.
At the beginning of the XX century and in the absence of advanced computational facilities, the simplifying physical models of line formation (e.g. pure absorption – LTE) had to be used. As the spectral lines are formed in the upper, low density regions of stellar atmospheres where radiative rates exceed the collisional ones and the scattering becomes important, in 1950’s the picture of spectrum formation was greatly improved when the assumption of local thermodynamic equilibrium (LTE) was abandoned in favor of statistical equilibrium or the so- called non-LTE (NLTE). The intrinsic difficulty of the NLTE line transfer problem is in the non- local and generally non-linear coupling between the atomic level populations and the radiation field intensities in the corresponding line transitions. Scattering process makes the properties of very distant points in a medium non-locally coupled through the radiation field. The specific intensity of the radiation field at a given depth point depends, via the RT process, on the physical properties over a wide range of distant points. On the other hand, the physical state of the medium at each point depends, via radiative excitations and ionizations, on the radiation field itself. The dependence of the atomic level populations on the radiation field intensity is expressed by the statistical equilibrium (SE) or rate equations describing the balance between all the transitions (radiative + collisional) that populate and those that depopulate each given atomic level. From the mathematical point of view the coupling of the radiation field and the excitation state of the gas is described by the simultaneous solution of the corresponding equations of radiative transfer and statistical equilibrium.
If the line source function (the ratio of line emissivity to opacity) can be explicitly written in terms of the mean line radiation field intensity, as in the case of a two-level atom line transfer, the problem is linear and can be solved by using either direct or iterative methods. However, in the general case of a multi-level atom line transfer, the problem is highly non-linear and an iterative solution must be sought in order to solve the problem.
In what follows we shall first discuss the basic equations of the multilevel line formation problem (Sec.2), then the methods nowadays in use for its solution (Sec.3) and, finally, two such efficient, simple and fast convergent iterative methods that do not require any matrix formalism – Iteration Factors Method and Forth-and-Back Implicit Lambda Iteration (Sec.4 and 5) and, consequently, exhibit advantages in terms of stability, as well as considerable savings of both computational time and memory storage.
2. THE MULTI-LEVEL ATOM LINE FORMATION PROBLEM
For the sake of an easier presentation of the methods, here we shall consider the schematic case of a multi-level atom line formation in a plane-parallel, stationary medium of constant properties with complete redistribution and no background continuum. The three-level hydrogen atom is an ideal test case for checking the numerical accuracy and convergence properties of various algorithms developed to solve the RT problem, and its solution has been obtained by many authors (see e.g. [1]). Under the above assumptions, the RT equation for any line transition ij between two discrete energy levels i and j ( ij EE > ) takes the form
23
O. Atanackovi: On two extremely fast methods for the solution of the non-lte line formation proble
)]()([)( τττ τ
−= , (1)
where Iνµ (τ) is the specific intensity of the radiation field at the mean line optical depth τ, ν is the frequency and µ is the cosine of the angle between the direction of photon propagation and the outward normal. The absorption line profile ν is assumed to be independent of both τ and µ. The line source function S contains the details of interactions between radiation and the atmospheric gas and depends on the populations of two bound atomic levels, i and j, the line transition is occurring between:
3
2
n A hS gnn B n B c n g
ν = =
− − . (2)
{ ( ) ( )}
( ) ( )
ij jk i ij ij k kj kj kj
n A B J C B J C n B J C n A B J C
φ φ
φ φ
Σ + + + Σ +
= Σ + + Σ + + (3)
with ji < , jk > . Here, C’s are the inelastic collisional rates, while radiative rates contain the so-called scattering integrals
1
1
1( ) ( ) ( ) 2
∞ ∞
−∞ −∞ −
= =∫ ∫ ∫ (4)
that account for the angle and frequency coupling of the specific intensities at the given depth point. For the NL-level atomic model, NL-1 linearly independent SE equations are closed by the particle conservation equation:
i tot i
n n=∑ (5)
where totn is the total number density of atoms.
3. NUMERICAL METHODS FOR THE SOLUTION OF MULTI-LEVEL PROBLEM
The above non-linear problem requires a simultaneous solution of the RT and SE equations, which can be achieved only through iterations. Various methods differ in the way linearity is achieved at each iterative step. The first robust method that could handle mutual non-
24
linear coupling of all relevant variables in a fully self-consistent manner was the complete linearization (CL) method by Auer and Mihalas [2]. The coupled equations are linearized in all variables and solved simultaneously for the variables’ corrections. The convergence rate of the CL method is very high. However, as a very large number of frequency, angle and depth points is usually needed for a good description of the physical system, huge memory storage and a lot of computing time are required. Moreover, the optimization of this global method is rather difficult.
Much simpler and cheaper methods were needed to solve more realistic and complex problems. These are the methods that use a sequential iterative procedure within the so-called structural approach (Simonneau and Crivellari [3]). They take advantage of the fact that a problem can be divided into separate tasks according to the physics of the problem and the mathematical properties of the relevant equations. The current solution of each individual task is obtained by assuming that the output of the others is known. Thus each task can be optimized independently. More specifically, in the multilevel line formation problem, each step of the iterative procedure is usually split in two parts so that within each of them, one of the two systems of equations, RT and SE, is solved by taking the solution, or part of the solution, of the other one as known. The coupling is performed at the end of the iterative procedure, with the convergence rate that depends on the amount of information transferred between the two parts.
The simplest and the well-known sequential iterative procedure is iteration that solves the problem equations in turn. However, in scattering-dominated media of large optical thickness (typical non-LTE conditions), the convergence of iteration is extremely slow. It follows the process of a particular scattering event and at each iteration step it corrects the solutions only within a unit optical path. It transfers from one part of the iterative step to the other more information than necessary.
There are two broad classes of methods attempting to facilitate the solution of NLTE line transfer problem and improving upon the difficulties encountered by the above mentioned methods. One is based on the so-called ETLA (Equivalent-Two-Level-Atom) approach, described in [4], in which only one transition in the atomic model at a time is combined with the transfer problem, whereas the coupling of all levels is achieved by iteration over all transitions. However, it may happen that iterative procedure converges upon inconsistent solutions as the multilevel transition interlocking is treated iteratively. The other approach employs an approximate treatment of RT coupled with the full set of SE equations. The exact operator is replaced by an approximate one, whereas a small error introduced by this approximation is accounted for iteratively. The approximation is based either on the physics of the transport of photons through the medium (escape probability, core saturation etc.) or on some computational considerations. Such approach is known as ALI (Approximated/Accelerated Lambda Iteration). Rybicki [5] and Cannon [6] pioneered the first ideas of ALI methods. Rybicki eliminated the cause of the slow convergence of simple iteration, i.e. poor conditioning of the equations arising from a large number of scatterings, by the elimination of scatterings in the line core (core saturation assumption) which do not contribute much to RT. The solution of the preconditioned equations that contain only wing components by direct iteration converged more rapidly. Cannon was the
25
first to use the idea of 'operator splitting' from the numerical analysis in radiative transfer computations. He replaced the full (exact) operator by the so called approximate Lambda operator - ALO and computed a small error term made by this approximation using a perturbation technique. His ALO was operator evaluated with fewer angular-frequency quadrature points.
ALO has to be simpler to invert than the operator, and also as close physical approximation of the exact operator as possible to ensure stable and rapid convergence to the exact solution. Olson et al. [7] were the first to point out that the diagonal (local) part of the exact matrix represents nearly optimum approximate operator. The most commonly used ALI method is the Jacobi iteration scheme that employs the diagonal ALO and computes the error introduced by this approximation iteratively. Using a 'short characteristic' solution of the first-order differential RT equation and parabolic approximation for the source function Olson and Kunasz [8] derived a fast method to compute the exact diagonal of the matrix. The approach that introduces ALO directly into SE equations to make them linear from the outset (preconditioning) is proposed by Rybicki and Hummer [9]. This is the well-known MALI (Multilevel Accelerated Lambda Iteration) method, which was successfully applied to the solution of various RT problems. The convergence rate of the Jacobi method is usually increased by the Ng acceleration technique [10]. The diagonal ALO is also used as the local-operator option of the well-known multilevel transfer code MULTI [11].
Another numerical method for NLTE RT applications that is twice as fast as the Jacobi method is the Gauss-Seidel (GS) method by Trujillo Bueno and Fabiani Bendicho [12]. Its implementation into multilevel RT is made explicit by Paletou and Leger [13]. Successive Overrelaxation (SOR) is applied to speed up its convergence. The use of additional mathematical techniques is not straightforward. It usually needs some preliminary analyses in order to prevent the divergence of the whole procedure. Therefore, despite important progress made thanks to ALI methods, the search for more efficient algorithms is still an open field of research.
Another approach that uses the so-called iteration factors, either in explicit or implicit way, is implemented in two extremely fast methods: the Iteration Factors Method (IFM) and the Forth-and-Back Implicit Lambda Iteration (FBILI), respectively. These are the methods developed for the NLTE line transfer problem by Atanackovic-Vukmanovic [14], Atanackovic- Vukmanovic and Simonneau [15], Atanackovic-Vukmanovic, Crivellari and Simonneau [16] and Kuzmanovska-Barandovska and Atanackovic [17]. They will be discussed here in more details.
4. ITERATION FACTORS METHOD
As we have already emphasized, a global astrophysical problem can be divided into several specific physical key-problems represented by individual blocks of a general block- diagram. The optimum structure of the block-diagram is determined by the physics itself. A current solution of each block is obtained assuming that the output from all the others is known. The solution of the global problem is thus achieved through a sequential iterative procedure. The most straightforward iteration scheme is very slow. However, its slight modification can lead to an extremely high convergence rate. One such revision is in the use of the so-called iteration
26
factors (IFs) as the input/output of the blocks that constitute the block diagram. At each iteration step the IFs are computed from the current solution and then used to update it. In order to provide fast convergence they have to be good quasi-invariants (to change little from one iteration to another). Because being defined as the ratios of the homologous physical quantities, they become exact already in the first few iterations leading quickly to the exact solution of the whole procedure.
In RT literature, the idea of iteration factors appeared for the first time in the paper by Feautrier [18] who suggested that the use of the ratio of two moments of the radiation field intensity could speed up the stellar atmosphere model computations. The first realization of the idea was the variable (depth-dependent) Eddington factor (VEF) technique [19] developed for the solution of the monochromatic transfer problem in plane-parallel geometry (Fig.1). Auer and Mihalas iterated on the ratio of the third to the first angular moments of the radiation field F=K/J, i.e. on the so-called Eddington factor.
Figure 1. Flow-chart of the method of VEFs for the monochromatic transfer problem.
They pointed out that it is much better to iterate on the ratio of two quantities than on the quantities themselves, as their ratio changes much less from one iteration to another. An extremely fast convergence is achieved at the cost of a small additional effort, with respect to the iteration, of iterative computation of Eddington factor and the solution of one second-order differential equation (i.e. two first-order differential equations closed by the iteration factor, see Fig.1). Two to three iterations were enough to reach the exact solution.
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The VEFs have been applied within the linearization method for both, the stellar atmosphere modeling [2] and the line formation [20], problems to reduce the angular dimensions of the system under study. They enabled a complex explicit frequency-angle coupling to be simplified, so that only the frequency coupling is treated explicitly, whereas the angle coupling is treated iteratively. However, a very large number of frequency points still made the computations with the CL/VEF demanding.
The idea of VEFs is generalized in the Iteration Factors Method (IFM), developed to solve the two-level-atom line transfer problem by Atanackovic-Vukmanovic [14] and Atanackovic-Vukmanovic and Simonneau [15]. For the first time they used both angle and frequency averaged iteration factors, defined for a line as a whole. At each iteration, depth- dependent factors defined as the ratios of the relevant (angle and frequency) intensity moments are computed from the current values of the radiation field and then used to close the system of the RT equation moments. The iteration factors method does not need any matrix operation, thus requiring small memory storage and computational time.
The IFM is extended to the solution of the multilevel line transfer problem by Kuzmanovska-Barandovska and Atanackovic [17]. Iterative procedure that overcomes the above mentioned drawback of the CL/VEF method is proposed, retaining its favorable convergence properties while becoming computationally cheap owing to the use of iteration factors defined for each line as a whole. The algorithm is displayed in Fig.2.
Figure 1. Flow-chart of the IFM for the line transfer problem. The algorithm is divided by dashed line in two parts that can be optimized independently.
In the first part of each iteration, starting from a given source function ( 0S ) or the one known from the previous iteration step ( iS ) we formally solve the RT equation (1) for each line
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transition. The obtained intensities are used for the computation of the relevant angle and frequency integrated moments and the iteration factors as their ratios. The IFs are then used as the known coefficients to close the system of the RT equation moments (for all line transitions) non-linearly coupled with the statistical equilibrium (SE) equations (for all level populations) and to update the solution in the second part of each iteration. Two approaches were used to solve these two systems of equations simultaneously. The first one is based on the linearization of the equations by neglecting the second- and higher-order terms. A substitution of the linearized SE equations into the linearized RT equation moments yields the system of equations that involve only the corrections of the frequency integrated mean intensity of the radiation field explicitly. In the second approach, the linear form of the SE equations is obtained by assuming that the level populations in their line-opacity-like terms are known from the previous iteration. Thus one derives linear relations between each line source function and the full set of the radiation field mean intensities and solves them together with the RT equation moments for all line transitions linearly coupled in this way. The second approach can be labeled "preconditioning". In both approaches an extremely fast convergence is provided and the cost per iteration drastically reduced by the use of quasi-invariant iteration factors in the linearly coupled RT equation moments and SE equations.
The proposed method is tested on a standard benchmark model for the multilevel atom problems: three-level hydrogen atoms in a plane-parallel, semi-infinite isothermal atmosphere with no background opacity. We compared the convergence rate of the IFM with that of the MALI method, using the results given in [9]. The convergence rate of the IFM is higher than that of MALI by a factor of more than 5 with respect to the solution with diagonal operator without acceleration (D/N), and by a factor of about 2 for the tridiagonal ALO without acceleration (T/N). Only the use of tridiagonal ALO with Ng acceleration (T/A) provided more rapid convergence than the IFM (that uses no additional acceleration). In Table 1. the convergence properties of the IFM are compared with those of the methods currently in use (MALI, Gauss-Seidel and SOR), the results of which for the same test problem are given by Paletou and Leger [13]. One can see that the total computational work (CW) of the IFM is 5-6 times lesser than that of MALI. IFM is about 4 times faster than GS and 1.5 times faster than GS accelerated by SOR.
Table 1: Comparison of the convergence properties of the IFM with other methods (MALI, GS and SOR [13]). Iter – number of iterations needed to satisfy a given cR (maximum relative change of the source function between two successive iterations), CPU/Iter – computational time per iteration and CW – total computational work.
Rc MALI GS SOR IFM Iter CPU/Iter CW Iter CPU/Iter
CW Iter CPU/Iter
CW Iter CPU/Iter
CW 10-2 100 0.163 16.3 55 0.21 11.6 25 0.21 5.25 12 0.238 2.86
10-3 170 0.163 27.7 85 0.21 17.9 35 0.21 7.35 19 0.238 4.52
10-4 230 0.163 37.5 120 0.21 25.2 0.21 8.40 26 0.238 6.19
Very high convergence rate is not affected by the refinement of the grid resolution, i.e. the number of iterations is practically insensitive to the spatial grid, so that the total computational
29
work scales linearly with the number of depth grid points (see Table 2). As an additional check the IFM was applied to the solution of a 5-level CaII ion model, the test problem proposed by Avrett [21], and similar, very encouraging results are obtained [22].
5. FORTH-AND-BACK IMPLICIT LAMBDA ITERATION
Another simple and efficient method developed to accelerate the convergence of the classical -iteration while retaining its straightforwardness is Forth-and-Back Implicit Lambda Iteration – FBILI [14, 16]. It makes use of the fact that even if the radiation field is not known, its physical behavior can be easily represented. An implicit representation of the source function is used in the computation of both the in-going ( −
νµI ) and the out-going ( + νµI ) intensities of the
radiation field that are considered separately within a forth-and-back approach. Namely, using the integral form of the RT equation
dtetSeII l
/)(/ 1 )()()( (6)
and assuming polynomial (e.g. piecewise parabolic) behavior for the source function in each depth subinterval τ , for every optical depth point lτ one can write implicit linear relations representing the specific intensities in terms of the yet-unknown values of the source function and its derivatives:
)(')()( lll ScSbaI τττ νµνµνµνµ ±±±± ++= (7)
According to the idea of iteration factors, the iterative computations of the coefficients of these implicit relations (implicit, as the source function is a priori unknown), rather than of the unknown functions themselves, greatly accelerate the convergence of the iterative scheme. The coefficients are computed layer by layer during the forward (‘in-going’) elimination process and stored for further use in the back substitution process when the explicit values of unknown functions are computed.
)(')() )(
where − νµa is to be computed with old (from the previous iteration) source function. Finally, the
coefficients of the linear relation for the unknown in-going mean radiation field
)(')()( lllll ScSbJ τττ −−− += , (9)
are stored. The aim is to obtain the coefficients of the linear relation between the 'full' mean intensity
and the source function ( bSaJ += ) which, when substituted into the statistical equilibrium equations, lead to the new source function. In order to obtain the 'full' coefficients a and b , the coefficients of the corresponding implicit relation for the out-going mean intensities are needed. Proceeding from the lower boundary condition we determine them simultaneously with the updated source function when sweeping layer by layer back to the surface. The process is iterated to convergence. A negligible additional computational effort made within FBILI method with respect to the classical iteration results in an extremely fast convergence to the exact solution. This is due to the fact that the only information transferred from the previous iteration step to the next one is in the ratio between the non-local part of the inward mean intensities and the current (old) source function ( 0/ Sa −
νµ ), which represents an iteration factor. The efficiency and the accuracy of the FBILI method have been verified on several RT
problems. The two-level-atom line transfer problem with complete redistribution (CRD) and its generalization to the case of partial frequency redistribution (PRD), as well as the multi-level- atom line formation problem with CRD, are considered by Atanackovi-Vukmanovi, Crivellari and Simonneau [16]. The exact solutions are obtained in a very small number of iterations even under extreme non-LTE conditions. The method is generalized to the monochromatic scattering and two-level atom line formation in static spherical media [23, 24], to the two-level line formation in media with low velocity fields [25] and to the two-level line formation in 2D Cartesian coordinates [26]. The multi-level atom line transfer in 1D is made more explicit in [27]. In Table 2, the comparison of the convergence properties of IFM and FBILI methods is shown. The CPU time per FBILI iteration is only 10% longer than CPU per iteration. In terms of the total computational work, FBILI is faster than IFM by a factor of 2. Both methods have higher convergence rate than those currently in use (see Tables 1 and 2). Another important advantage of the two methods with respect to the ALI ones is that their total CW scales linearly with the number of grid points. Table 2: Convergence properties of the IFM and FBILI methods: the number of iterations (Iter) required to satisfy the criterion 410−=cR , the computation time per iteration (CPU/Iter) in seconds and total computational work (CW) for the three-level H atom benchmark problem as a function of the number of depth points per decade τN .
τN iteration IFM FBILI
CPU/Iter Iter CPU/Iter CW Iter CPU/Iter CW
3 0.0135 23 0.0183 0.422 11 0.0156 0.172
6 0.0264 23 0.0353 0.813 15 0.0292 0.438
9 0.0395 28 0.0525 1.469 0.0436 0.828
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6. CONCLUSIONS
In order to speed up the convergence, the iterative methods nowadays in use to solve various RT problems usually employ some additional mathematical acceleration techniques, the implementation of which is not straightforward and may lead to divergence of the iterative procedure. We have shown that the use of iteration factors (either explicitly – Iteration Factors Method, or implicitly – Forth-and-Back Implicit Lambda Iteration) enables an extremely fast convergence to the exact solution of NLTE radiative transfer problems with no need for an extra acceleration.
In particular, the advantages of the IFM with respect to the CL and ALI methods are as follows: (1) the iteration factors defined for line as a whole reduce drastically the need for large memory and computing time and (2) there is no increase in the number of iterations with the grid resolution refinement. The FBILI method is even more simple, stable and extremely fast. The facts that it is a two-point algorithm and that the memory storage grows linearly with the dimensions of the problem makes FBILI very suitable for 3D RT.
Both methods are rapidly converging, exact and stable, being thus very promising in the applications to more complex problems, in which RT is coupled with other physical phenomena.
ACKNOWLEDGEMENTS
This work has been realized within the project No. 176004 (”Stellar Physics”) supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia.
REFERENCES
[1] E.H. Avrett and R. Loeser, in Numerical Radiative Transfer, ed. W. Kalkofen (Cambridge Univ. Press), p. 135 (1987)
[2] L.H. Auer and D. Mihalas, Astrophys. J. 158, 641-655 (1969) [3] E. Simonneau and L. Crivellari, Structural algorithms to solve radiative transfer problems,
Research Project, Instituto de Astrofisica de Canarias (1994) [4] D. Mihalas, Stellar Atmospheres, 2nd edition, Freeman, San Francisco (1978) [5] G. Rybicki, in: R.G. Athay, L.L.House, G.Jr Newkirk, editors. Line formation in the
presence of magnetic fields. High Altitude Observatory, Boulder; p.145 (1972) [6] C.J. Cannon, Journal of Quant. Spectrosc. Radiat. Transfer 13, 627-633 (1973) [7] G.L. Olson, L.H. Auer, J.R. Buchler, Journal of Quant. Spectrosc. Radiat. Transfer 35, 431-
442 (1986) [8] G.L. Olson and P.B. Kunasz, Journal of Quant. Spectrosc. Radiat. Transfer 38, 325-336
(1987) [9] G. Rybicki and D. Hummer, Astron. Astrophys. 245, 171-181 (1991) [10] K.C. Ng, J. Chem. Phys., 61, 2680-2689 (1974) [11] M. Carlsson, A Computer Program for Solving Multi-Level Non-LTE Radiative Transfer
Problems in Moving or Static Atmospheres, Uppsala Astron. Obs. Rep. 33, (1986, 1991) [12] J. Trujillo Bueno, P. Fabiani Bendicho, Astrophys. J. 455, 646-657 (1995) [13] F. Paletou, L. Leger, Journal of Quant. Spectrosc. Radiat. Transfer 103, 57-66 (2007) [14] O. Atanackovi-Vukmanovi, PhD thesis, Department of Astronomy, Faculty of
Mathematics, University of Belgrade (1991)
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[15] O. Atanackovi-Vukmanovi, E. Simonneau, Journal of Quant. Spectrosc. Radiat. Transfer 51, 525, (1994)
[16] O. Atanackovi-Vukmanovi, L. Crivellari and E. Simonneau, Astrophys. J. 487, 735-746 (1997)
[17] O. Kuzmanovska-Barandovska and O. Atanackovi, Journal of Quant. Spectrosc. Radiat. Transfer 111, 708-722 (2010)
[18] P.Feautrier, C.R.Acad.Sci., Paris, 258, p. 3189 (1964) [19] L.H. Auer, D. Mihalas, MNRAS 149, 65 (1970) [20] L. Auer, Astrophys. J. 180, 469-472 (1973) [21] E.H. Avrett, Resonance Lines in Astrophysics, National Center for Atmospheric Research,
Boulder, 27 (1968) [22] O. Kuzmanovska-Barandovska and O. Atanackovi, Publ. Astron. Obs. Belgrade 90, 155-
158 (2010) [23] O. Atanackovi-Vukmanovi, Serb. Astron. J. 167, 27-34 (2003) [24] O. Atanackovi-Vukmanovi, ASP Conference Series 370, 97-102 (2007) [25] I. Pirkovi and O. Atanackovi, Serb. Astron. J. 189, 53-67 (2014) [26] I. Mili and O. Atanackovi, Advances in Space Research 54, 1297-1307 (2014) [27] O. Kuzmanovska, O.Atanackovi and M. Faurobert, in preparation (2016)
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METHOD OF ABSOLUTE GRAVITY MEASUREMENT AND ITS INFLUENCE IN MASS DETERMINATION
Luljeta Disha1, Defrim Bulku1 and Antoneta Deda2
1General Directorate of Metrology, Autostrada Tr-Dr, km 8, Kashar,1000 Tirana, Albania
2 Faculty of Natural Science, Bulevardi Zogu I, 1000 Tirana, Albania
Abstract. The determination of acceleration due to gravity has application in many measurements. Its value is of interest in a broad area of physical sciences, Metrology, Geophysics and Geodesy. In Metrology, knowledge of the gravity field is very important in mass determination, and also it is used as a transfer standard in the determination of metrological standards involving force and pressure measurements. The need for modern absolute gravity measurements within 10-9 relative uncertai

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